The Central Limit Theorem is one of the greatest results in probability theory because it says that the sum of a big number of variables has approximately a normal distribution.
We define a set of random variables iid X1, X2, X3, … , Xn with mean μ and variance σ². Then if n is very big, the sum
X1 + X2 + X3 + … + Xn
is approximately normal with mean nμ and variance nσ². If we also normalize the sum we can say that
(X1 + X2 + X3 + … + Xn – nμ)/(σ√n)
is approximately a normal standard, so to a normal with mean 0 and variance 1.
Look for the most popular distributions in statistics
Distribution can be divided into two categories: discrete and continuous distributions.
The most popular discrete distributions are:
1) Boolean (Bernoulli) which takes value 1 with probability p and value 0 with probability q = 1 − p.
2) Binomial which describes the number of successes in a series of independent Yes/No experiments all with the same probability of success
3) Poisson which describes a very large number of individually unlikely events that happen in a certain time interval
4) Hypergeometric which describes the number of successes in the first m of a series of n consecutive Yes/No experiments, if the total number of successes is known. This distribution arises when there is no replacement.
The most popular continuous distributions are:
1) Normal (or Gaussian) often used in the natural and social sciences to represent real-valued random variables
2) Chi-squared which is the sum of the squares of n independent Gaussian random variables. It is a special case of the Gamma distribution
3) Gamma which describes the time until n consecutive rare random events occur in a process with no memory
4) Beta a family of two-parameter distributions with one mode, of which the uniform distribution is a special case, and which is useful in estimating success probabilities
5) T-Student useful for estimating unknown means of Gaussian populations
6) F-Distribution (Fisher) is a continuous probability distribution that arises frequently as the null distribution of a test statistic, most notably in the analysis of variance
7) Weibull of which the exponential distribution is a special case is used to model the lifetime of technical devices and is used to describe the particle size distribution of particles generated by grinding, milling and crushing operations